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21: 10.54 Integral Representations

§10.54 Integral Representations

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10.54.2 $${𝗃}_n\left(z\right)=\frac{{(-\mathrm{i})}^n}{2}{\int }_0^{\pi }{\mathrm{e}}^{\mathrm{i}z\cos \theta }{P}_n\left(\cos \theta \right)\sin \theta d\theta .$$

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Symbols:

$P_n\left(x\right)$: Legendre polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $\sin z$: sine function, ${𝗃}_n\left(z\right)$: spherical Bessel function of the first kind, $n$: integer and $z$: complex variable

A&S Ref:

10.1.14

Permalink:

http://dlmf.nist.gov/10.54.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.54 and Ch.10

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10.54.3 $${𝗄}_n\left(z\right)=\frac{\pi }{2}{\int }_1^{\mathrm{\infty }}{\mathrm{e}}^{-zt}{P}_n\left(t\right)dt,$$

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Symbols:

$P_n\left(x\right)$: Legendre polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{ph}$: phase, ${𝗄}_n\left(z\right)$: modified spherical Bessel function, $n$: integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/10.54.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.54 and Ch.10

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${𝗁}_n^{(1)}\left(z\right)={\displaystyle \frac{{(-\mathrm{i})}^{n+1}}{\pi }}{\displaystyle {\int }_{\mathrm{i}\mathrm{\infty }}^{(1+)}}{\mathrm{e}}^{\mathrm{i}zt}{Q}_n\left(t\right)dt,$

… ►For the Legendre polynomial $P_n$ and the associated Legendre function $Q_n$ see §§18.3 and 14.21(i), with $\mu =0$ and $\nu =n$. …

22: 36.5 Stokes Sets

… ►For $z\ne 0$, the Stokes set is expressed in terms of scaled coordinates … ►

36.5.7 $$X=\frac{9}{20}+20u^4-\frac{Y^2}{20u^2}+6u^2\mathrm{sign}\left(z\right),$$

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Symbols:

$\mathrm{sign}x$: sign of, $z$: real parameter, $X$: scaled coordinate and $Y$: scaled coordinate

Permalink:

http://dlmf.nist.gov/36.5.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §36.5(ii), §36.5(ii), §36.5 and Ch.36

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36.5.10 $$160u^6+40u^4=Y^2.$$

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Symbols:

$Y$: scaled coordinate

Permalink:

http://dlmf.nist.gov/36.5.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §36.5(ii), §36.5(ii), §36.5 and Ch.36

… ►With coordinates … ►

36.5.17 $$Y_{\mathrm{S}}(X)=Y(u,|X|),$$

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Symbols:

$X$: scaled coordinate

Permalink:

http://dlmf.nist.gov/36.5.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §36.5(iii), §36.5(iii), §36.5 and Ch.36

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23: 6.10 Other Series Expansions

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§6.10(ii) Expansions in Series of Spherical Bessel Functions

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6.10.4 $$\mathrm{Si}\left(z\right)=z\sum _{n=0}^{\mathrm{\infty }}{\left({𝗃}_n\left(\frac{1}{2}z\right)\right)}^2,$$

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Symbols:

$\mathrm{Si}\left(z\right)$: sine integral, ${𝗃}_n\left(z\right)$: spherical Bessel function of the first kind, $z$: complex variable and $n$: nonnegative integer

Referenced by:

§6.10(ii)

Permalink:

http://dlmf.nist.gov/6.10.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §6.10(ii), §6.10 and Ch.6

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6.10.5 $$\mathrm{Cin}\left(z\right)=\sum _{n=1}^{\mathrm{\infty }}{a}_n{\left({𝗃}_n\left(\frac{1}{2}z\right)\right)}^2,$$

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Symbols:

$\mathrm{Cin}\left(z\right)$: cosine integral, ${𝗃}_n\left(z\right)$: spherical Bessel function of the first kind, $z$: complex variable, $n$: nonnegative integer and $a_n$: coefficients

Permalink:

http://dlmf.nist.gov/6.10.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §6.10(ii), §6.10 and Ch.6

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6.10.6 $$\mathrm{Ei}\left(x\right)=\gamma +\ln \left|x\right|+\sum _{n=0}^{\mathrm{\infty }}{(-1)}^n(x-a_n){\left({𝗂}_n^{(1)}\left(\frac{1}{2}x\right)\right)}^2,$$ $x\ne 0$,

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Symbols:

$\gamma$: Euler’s constant, $\mathrm{Ei}\left(x\right)$: exponential integral, $\ln z$: principal branch of logarithm function, ${𝗂}_n^{(1)}\left(z\right)$: modified spherical Bessel function, $x$: real variable, $n$: nonnegative integer and $a_n$: coefficients

Referenced by:

§6.18(i)

Permalink:

http://dlmf.nist.gov/6.10.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §6.10(ii), §6.10 and Ch.6

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6.10.8 $$\mathrm{Ein}\left(z\right)=z{\mathrm{e}}^{-z/2}\left({𝗂}_0^{(1)}\left(\frac{1}{2}z\right)+\sum _{n=1}^{\mathrm{\infty }}\frac{2n+1}{n(n+1)}{𝗂}_n^{(1)}\left(\frac{1}{2}z\right)\right).$$

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Symbols:

$\mathrm{Ein}\left(z\right)$: complementary exponential integral, $\mathrm{e}$: base of natural logarithm, ${𝗂}_n^{(1)}\left(z\right)$: modified spherical Bessel function, $z$: complex variable and $n$: nonnegative integer

Referenced by:

§6.10(ii)

Permalink:

http://dlmf.nist.gov/6.10.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §6.10(ii), §6.10 and Ch.6

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24: Bibliography M

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Spherical Harmonics. An Elementary Treatise on Harmonic Functions with Applications. 3rd edition, International Series of Monographs in Pure and Applied Mathematics, Vol. 98, Pergamon Press, Oxford.

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External Links:

MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

Ch.14.

Encodings:

BibTeX

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On the evaluation of the integral over the product of two spherical Bessel functions. J. Math. Phys. 32 (3), pp. 642–648.

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External Links:

ISSN 0022-2488, Document, MathReview (S. K. Chatterjea), ZentralBlatt

Referenced by:

§1.17(ii), §10.59.

Encodings:

BibTeX

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Analytic expressions for integrals of products of spherical Bessel functions. J. Phys. A 24 (7), pp. 1435–1453.

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External Links:

ISSN 0305-4470, Document, MathReview, ZentralBlatt

Referenced by:

§10.59.

Encodings:

BibTeX

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Lie theory and separation of variables. I: Parabolic cylinder coordinates. SIAM J. Math. Anal. 5 (4), pp. 626–643.

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External Links:

Document, MathReview (G. R. Allcock), ZentralBlatt

Referenced by:

§12.13(ii), §12.17.

Encodings:

BibTeX

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Field Theory Handbook. Including Coordinate Systems, Differential Equations and Their Solutions. 2nd edition, Springer-Verlag, Berlin.

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Notes:

Reprinted in 1988.

External Links:

ISBN 3-540-18430-9, MathReview

Referenced by:

Table 28.1.1.

Encodings:

BibTeX

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25: 22.18 Mathematical Applications

… ►In polar coordinates, $x=r\cos \varphi$, $y=r\sin \varphi$, the lemniscate is given by $r^2=\cos \left(2\varphi \right)$, $0\le \varphi \le 2\pi$. … ►Discussion of parametrization of the angles of spherical trigonometry in terms of Jacobian elliptic functions is given in Greenhill (1959, p. 131) and Lawden (1989, §4.4). …

26: 30.2 Differential Equations

… ►In applications involving prolate spheroidal coordinates ${\gamma }^2$ is positive, in applications involving oblate spheroidal coordinates ${\gamma }^2$ is negative; see §§30.13, 30.14. … ►If $\gamma =0$, Equation (30.2.4) is satisfied by spherical Bessel functions; see (10.47.1).

27: 10.59 Integrals

§10.59 Integrals

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10.59.1

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Symbols:

$P_n\left(x\right)$: Legendre polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, ${𝗃}_n\left(z\right)$: spherical Bessel function of the first kind and $n$: integer

A&S Ref:

11.4.26

Referenced by:

§10.59

Permalink:

http://dlmf.nist.gov/10.59.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.59 and Ch.10

►where $P_n$ is the Legendre polynomial (§18.3). ►For an integral representation of the Dirac delta in terms of a product of spherical Bessel functions of the first kind see §1.17(ii), and for a generalization see Maximon (1991). …

28: 30.10 Series and Integrals

… ►For expansions in products of spherical Bessel functions, see Flammer (1957, Chapter 6).

29: 7.6 Series Expansions

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§7.6(ii) Expansions in Series of Spherical Bessel Functions

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7.6.8 $$\mathrm{erf}z=\frac{2z}{\sqrt{\pi }}\sum _{n=0}^{\mathrm{\infty }}{(-1)}^n\left({𝗂}_{2n}^{(1)}\left(z^2\right)-{𝗂}_{2n+1}^{(1)}\left(z^2\right)\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erf}z$: error function, ${𝗂}_n^{(1)}\left(z\right)$: modified spherical Bessel function, $z$: complex variable and $n$: nonnegative integer

A&S Ref:

7.1.7 (in different form)

Referenced by:

§7.6(ii)

Permalink:

http://dlmf.nist.gov/7.6.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §7.6(ii), §7.6 and Ch.7

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7.6.9 $$\mathrm{erf}\left(az\right)=\frac{2z}{\sqrt{\pi }}{\mathrm{e}}^{(\frac{1}{2}-a^2)z^2}\sum _{n=0}^{\mathrm{\infty }}{T}_{2n+1}\left(a\right){𝗂}_n^{(1)}\left(\frac{1}{2}{z}^2\right),$$ $-1\le a\le 1$.

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Symbols:

$T_n\left(x\right)$: Chebyshev polynomial of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erf}z$: error function, $\mathrm{e}$: base of natural logarithm, ${𝗂}_n^{(1)}\left(z\right)$: modified spherical Bessel function, $z$: complex variable and $n$: nonnegative integer

Referenced by:

§7.6(ii)

Permalink:

http://dlmf.nist.gov/7.6.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §7.6(ii), §7.6 and Ch.7

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7.6.10 $$C\left(z\right)=z\sum _{n=0}^{\mathrm{\infty }}{𝗃}_{2n}\left(\frac{1}{2}\pi z^2\right),$$

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Symbols:

$C\left(z\right)$: Fresnel integral, $\pi$: the ratio of the circumference of a circle to its diameter, ${𝗃}_n\left(z\right)$: spherical Bessel function of the first kind, $z$: complex variable and $n$: nonnegative integer

Referenced by:

§7.6(ii)

Permalink:

http://dlmf.nist.gov/7.6.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §7.6(ii), §7.6 and Ch.7

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7.6.11 $$S\left(z\right)=z\sum _{n=0}^{\mathrm{\infty }}{𝗃}_{2n+1}\left(\frac{1}{2}\pi z^2\right).$$

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Symbols:

$S\left(z\right)$: Fresnel integral, $\pi$: the ratio of the circumference of a circle to its diameter, ${𝗃}_n\left(z\right)$: spherical Bessel function of the first kind, $z$: complex variable and $n$: nonnegative integer

Referenced by:

§7.6(ii)

Permalink:

http://dlmf.nist.gov/7.6.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §7.6(ii), §7.6 and Ch.7

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30: 20 Theta Functions

Chapter 20 Theta Functions

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